Title: Change detection and inference in high-dimensional covariance matrices based on $l_1$- and $l_2$-projections
Authors: Ansgar Steland - University Aachen (Germany) [presenting]
Rainer von Sachs - Université catholique de Louvain (Belgium)
Abstract: New results about inference and change point analysis of high dimensional vector time series are discussed. The results deal with change-point procedures that can be based on an increasing number of bilinear forms of the sample variance-covariance matrix as arising, for instance, when studying change-in-variance problems for projection statistics. Areas of applications are shrinkage, aggregated sensor data and dictionary learning. Contrary to many known results, e.g. from random matrix theory, the results hold true without any constraint on the dimension, the sample size or their ratio, provided the weighting vectors are uniformly $l_1$-bounded. Extensions to a large class of $l_2$-bounded projections are also discussed. The large sample approximations are in terms of (strong resp. weak) approximations by Gaussian processes for partial sum and CUSUM type processes. It turns out that the approximations by Gaussian processes hold not only without any constraint on the dimension, the sample size or their ratios, but even without any such constraint with respect to the number of bilinear form. For the unknown variances and covariances of these bilinear forms nonparametric estimators are proposed and shown to be uniformly consistent. We also discuss how the theoretical results lead to novel distributional approximations and sequential methods for shrinkage covariance matrix estimators.